Class 9 Math: Circles Quick Revision Chapter Summary

Need to review the entire chapter before a test? This quick revision guide summarizes all the core concepts, topics, and equations taught in Class 9 Circles to help you refresh your memory instantly.

Core Topics At a Glance

### 📌 Angle subtended by chord at a point A chord subtends equal angles at equal distances. If two chords of a circle are equal, they subtend equal angles at the center of the circle. Conversely, if chords subtend equal angles at the center, the chords are equal.

### 📌 Perpendicular from center to a chord A perpendicular line drawn from the center of a circle to a chord bisects the chord. Conversely, the line joining the center to the midpoint of a chord is perpendicular to the chord.

### 📌 Equal chords and distances from center Equal chords of a circle are equidistant from the center. Conversely, chords that are equidistant from the center of a circle are equal in length.

### 📌 Angle subtended by an arc of a circle The angle subtended by an arc at the center of a circle is double the angle subtended by it at any point on the remaining part of the circle. A corollary is that angles in the same segment of a circle are equal.

### 📌 Cyclic quadrilaterals properties A quadrilateral is called cyclic if all its four vertices lie on a circle. The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees. Conversely, if the sum of opposite angles is 180°, the quadrilateral is cyclic.

Key Formulas Mind Map

• Double Angle Theorem: `\theta_{center} = 2 \theta_{boundary}` - Angle subtended by arc at center is twice angle at circumference....
• Cyclic Quadrilateral interior sum: `A + C = 180^\circ \land B + D = 180^\circ` - Opposite angles of cyclic quadrilaterals are supplementary....

Revision Checklist

1. Review the subtopic guides and outline notes.
2. Practice writing the key formulas from memory.
3. Solve at least 3 worked examples and check your steps.
4. Take a timed practice quiz to verify your speed.